Solve for $x$ and $y$ using elimination. $\begin{align*}-8x-2y &= -1 \\ 9x+y &= 6\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $1$ and the bottom equation by $2$ $\begin{align*}-8x-2y &= -1\\ 18x+2y &= 12\end{align*}$ Add the top and bottom equations. $10x = 11$ Divide both sides by $10$ and reduce as necessary. $x = \dfrac{11}{10}$ Substitute $\dfrac{11}{10}$ for $x$ in the top equation. $-8( \dfrac{11}{10})-2y = -1$ $-\dfrac{44}{5}-2y = -1$ $-2y = \dfrac{39}{5}$ $y = -\dfrac{39}{10}$ The solution is $\enspace x = \dfrac{11}{10}, \enspace y = -\dfrac{39}{10}$.